I misread the falconer card. In this case, I can play falconer because of the card gained by monkey, not directly because of the card my opponent gain.....
Monkey does not gain you any cards, Monkey allows you to draw cards you already have in your deck. In your original example, if you draw a Falconer and the initial gain by opponent had multiple types, you can play it reacting to your opponent's gain.
I also want to ask, can I play multiple reaction cards on ONE card that my opponent plays? Say he plays witch, I use guard dog to draw 2cards, I draw a moat, can I use the moat to block the witch attack?
This seems to be the core of your misunderstanding. You can play as many applicable Reactions as you have available, one after the other. In an obviously extreme example, if you have a hand of 10 Guard Dogs and your opponent plays an attack, you can play them all, drawing 20 cards alltogether. And if there is a Moat among them, you can play it afterwards to be unaffected by the attack.
I'm describing this in more detail below, in response to your other question. But the main thing is that there can be many Reactions and/or triggers that apply to a single gain (or other event).
So, if I have a monkey in play, and my opponent gains a province (victory card), and I draw a card (Black Cat), I cannot play it, because I only have Black Cat after my opponent finishes resolving the gain effect of the province. Is this right..?
No, that is incorrect. You can play the Black Cat. Your opponent gains the Province and puts it into their discard pile. Gaining the Province is over at this time. Now starts a "timing window" for Reactions/Triggers, where all players can use Reaction cards (like Black Cat) or existing triggered effects (like from Monkey) to do things. Only when all players have said they don't want to do anything regarding this any more, does that "timing window" close.
Your opponent gets first shot to do something (they are the active player), they likely have nothing to do. So now the "game asks you" if you have any, the answer is yes, a mandatory one, from Monkey. So you resolve it, and draw a card (we assume it's a Black Cat).
Since something happened, the game returns to the beginning and asks your opponent again if they want to react or use a trigger on their Province gain. Again, they don't have one, so the game asks you again, and again you have one (an optional one), you can play the Black Cat! You do that and draw 2 cards and your opponent gets a Curse.
Something happened, so again "the game asks" your opponent if they want to react or use a trigger on their Province gain. For the third time, they have nothiing to do, so the game asks you. Now, assuming you didn't draw another Black Cat, you finally have nothing to do.
Both you and your opponent declined to do anything, so only now is the time when the timing window closes that allows any of you to use a Reaction or Trigger on the Province gain.